Category of profinite groups

Question

My question is simple:

Is the category of profinite groups an accessible category?

Thank you

Edit: I will add the (hopefully simpler) question:

Is the category of profinite groups complete and does it admit a cogenerator?

Answer 1

This is a partial answere. Let $\mathcal{C}$ this category.

1) It's dual is finitely presentable: form profinite group this category is $Pro(Gr_f)$ where $Gr_f$ is the category of finite groups, for definition $Pro(Gr_f)=(Ind(Gr_f^{op}))^{op}$. Now, $Gr_f^{op}$ has finite colimits, then $Ind(Gr_f^{op})$ is equivalent to the category $Cont_f(Gr_f^{op}, Set)$ of functors $P: Gr_f^{op} \to Set$ such that $P^{op}$ preserving finite colimit, from T.1.46 of LPAC follow that $\mathcal{C}$ is finitely presentable.

2) From T.1.67 of LPAC follow that $\mathcal{C}$ cannot be locally presentable (isn't a complete lattice)

3) From c.2.47 of LPAC follow that $\mathcal{C}$ cannot be locally accessible and complete.

EDIT:

4) Now $Ind(Gr_f^{op})$ as any Ind-category (that is a finite cocompletion) has finite colimit, then $\mathcal{C}$ has finite limits. Furthermore $\mathcal{C}$ has (small) products ( as category of Hausdorff, compact, and totally disconnected, grops). THen $\mathcal{C}$ is complete.

THen from (3) $\mathcal{C}$ isn't locally accessible.

LPAC: LOCALLY PRESENTABLE AND ACCESSIBLE CATEGORIES. J.Adámek and J.Rosický.

Answer 2

The category of profinite groups does admit a cogenerator. Just take the product over all finite groups with the product topology. If $f, g : G \to H$ is a parallel pair of morphisms between profinite groups, then $f, g$ can be distinguished by a map out of $H$ into some finite group. This finite group in turn embeds into the product over all finite groups.

Answer 3

The category of profinite groups is complete and admits a small strong cogenerating family because its opposite is locally finitely presentable; indeed:

1. Any locally finitely presentable category is cocomplete and admits a small strong generating family.
2. The category of profinite groups is the pro-completion of the category of finite groups, which means its opposite is the ind-completion of the opposite of the category of finite groups; the latter has finite colimits, and as a general fact the ind-completion of a small category with finite colimits is a locally finitely presentable category.

Because the category of profinite groups is pointed, this is enough to imply that it has a cogenerator: just take the product of the small cogenerating family.

Incidentally, the argument I suggested in the comments here goes through now: the opposite of a locally finitely presentable category is locally finitely presentable if and only if it is a preorder; but a complete category is accessible if and only if it is locally presentable. These facts can be found in [Adámek and Rosický, Locally presentable and accessible categories], as cited in Buschi Sergio's answer. The fact that the category of profinite groups is the pro-completion of the category of finite groups is actually non-trivial and is shown in [Johnstone, Stone spaces].